TOMASZ BIGAJ THE INDISPENSABILITY ARGUMENT – A NEW CHANCE FOR EMPIRICISM IN MATHEMATICS? ABSTRACT. In recent years, the so-called indispensability argument has been given a lot of attention by philosophers of mathematics. This argument for the existence of mathematical objects makes use of the fact, neglected in classical schools of philosophy of mathematics, that mathematics is part of our best scientific theories, and therefore should receive similar support to these theories. However, this observation raises the question about the exact nature of the alleged connection between experience and mathematics (for example: is it possible to falsify empirically any mathematical theorems?). In my paper I would like to address this question by considering the explicit assumptions of different versions of the indispensability argument. My primary claim is that there are at least three distinct versions of the indispensability argument (and it can be even suggested that a fourth, separate version should be formulated). I will mainly concentrate my discussion on this variant of the argument, which suggests the possibility of empirical confirmation of mathematical theories. A large portion of my paper will focus on the recent discussion of this topic, starting from the paper by E. Sober, who in my opinion put reasonable requirements on what is to be counted as an empirical confirmation of a mathematical theory. I will develop his model into three separate scenarios of possible empirical confirmation of mathematics. Using an example of Hilbert space in quantum mechanical description I will show that the most promising scenario of empirical verification of mathematical theories has nevertheless untenable consequences. It will be hypothesized that the source of this untenability lies in a specific role which mathematical theories play in empirical science, and that what is subject to empirical verification is not the mathematics used, but the representability assumptions. Further I will undertake the problem of how to reconcile the alleged empirical verification of mathema- tics with scientific practice. I will refer to the polemics between P. Maddy and M. Resnik, pointing out certain ambiguities of their arguments whose source is partly the failure to distinguish carefully between different senses of the indis- pensability argument. For that reason typical arguments used in the discussion are not decisive, yet if we take into account some metalogical properties of applied mathematics, then the thesis that mathematics has strong links with experience seems to be highly improbable. KEY WORDS: applicability of mathematics, confirmation, empiricism, mathe- matical realism, quantum mechanics Foundations of Science 8: 173–200, 2003. © 2003 Kluwer Academic Publishers. Printed in the Netherlands.